Dissociation is a reversible process of splitting molecules dissolved in water drops into ions. It is treated as an equilibrium process and is described using the dissociation constants. The dissociation constant of chemical compound A is denoted here by KA. The dissociation constants are corrected for temperature using the formula of : KA(T)=KA(T0)exp⁡-ΔHdR1T-1T0, where ΔHD denotes the reaction enthalpy of dissociation at constant temperature and pressure, T is the temperature of air and R is the gas constant. The list of considered dissociation constants and their temperature dependence coefficients are available in Table . The dissociation of water, although very small, is also taken into account

The concentration of undissociated water molecules is so big that it is usually assumed constant and it traditionally multiplies the dissociation constant of water. This leads to a different definition of the dissociation constant for water: KH2O=[H+][OH-]

It is assumed that there is no electric charge in water drops and therefore the concentrations of positive and negative ions created during dissociation should balance each other. Using the dissociation constants (see Table ), all ion concentrations can be expressed as a function of the total concentration of the dissolved chemical compounds and the concentration of H+ ions. For an example derivation, see Sect. 7.6.2 in . The neutral charge condition can be expressed as [H+]+[N-III][H+]KNH3KH2O+KNH3[H+]︷positive ions=[NV]KHNO3[H+]+KHNO3+[SIV]KSO2([H+]+2KHSO3)([H+]2+[H+]KSO2+KSO2KHSO3)︷negative ions+KH2O[H+]+[SVI]([H+]+2KH2SO4)([H+]+KH2SO4)+[CIV]KCO2([H+]+2KHCO3)([H+]2+[H+]KCO2+KCO2KHCO3)︸negative ions.

The [] brackets denote the concentration of each of the chemical compounds (traditionally defined in units of moles per litre), capital letters denote the chemical compound and roman numbers mark its oxidation state. In Eq. () the dissociation constants of SO32-, CO32-, and SO42- ions (i.e. KHSO3, KHCO3, and KH2SO4) are multiplied by a factor of 2 to take into account the respective electric charge number of those ions.

Equation () has only one unknown variable – the new equilibrium concentration of the H+ ions. The new concentration is obtained iteratively using a numerical root-finding algorithm

TOMS 748 algorithm from Boost library. See https://www.boost.org/doc/ (last access: 28 August 2018) for documentation and for derivation.

When the SD wet radius is quickly changing, for example during the initial condensational growth of cloud droplet or rain drop evaporation, the dissociation procedure requires small time steps to reach convergence. The time step used in the dissociation procedure can be divided into a user-specified number of sub-steps in order to prevent limiting the overall simulation time step by dissociation.

The amount of chemical compound that can dissolve into a water drop from the gas phase is proportional to its partial pressure above the surface of the drop. Due to the longer timescale of the process, in contrast to dissociation, the transfer between the gas and liquid phases is not treated as an instantaneous process. Assuming that the water drop is internally mixed, the gas–liquid transfer is limited by the diffusion of gas-phase particles to the drop surface (gas-phase limitation) and the probability that the molecule will enter the drop after collision (interfacial limitation). Following chapter 8.4.2 in , for a chemical compound “A” the rate of transfer from the gas phase to the aqueous phase is given by d[A]dt=4rw3〈v〉αMA+rw23DA-1cA∞-[A]HAeffRT where DA and αMA are the diffusion and mass accommodation coefficients for the chemical compound “A”, 〈v〉 = 8RTπMA is the average velocity of the molecules calculated from the Maxwell–Boltzmann distribution function, MA is the molar mass of the chemical compound “A”, cA∞ is the ambient concentration of the trace gas “A”, and HAeff is the effective Henry's law constant of the chemical compound “A” (i.e. the equilibrium dissolution constant). The Henry's law constants depend on the temperature following a similar relation as for dissociation, Eq. (). Table  shows the Henry's law constants and their temperature dependencies and Table  presents the diffusion and mass accommodation coefficients. The term “effective” marks that the dissolution constants take into account the increase in the efficiency due to dissociation (see , Sect. 7.3 for the exact equations). Equation () is solved for each SD and for each of the considered trace gases. It is solved implicitly with respect to the aqueous-phase concentration and explicitly with respect to the gas-phase concentration. The input ambient trace gas concentration is calculated from the trace gas mixing ratio provided by the driver model to which the PBMC scheme is coupled. Obtained aqueous-phase concentration is recalculated to the mass of dissolved chemical compounds and the corresponding SD attribute is updated. The changes in the ambient trace gas mixing ratios are calculated by the PBMC scheme by summing the changes in chemical composition in all SDs in a given grid cell and then subtracting them from the trace gas mixing ratio of the driver model. To ensure that the sum of sinks from each SD does not exceed the available ambient trace gas mixing ratio, a relatively short time step should be applied. If necessary, the user can divide the model time step into sub-steps.

The reaction rates of oxidation by ozone and hydrogen peroxide can be described as a : RO3=d[SVI]dt|O3=k0+k1KSO2[H+]+k2KSO2KHSO3[H+]2[O3][SO2*H2O], RH2O2=d[SVI]dt|H2O2=k3KSO21+k4[H+][H2O2][SO2*H2O], where RA is the reaction rate of the chemical compound “A” and k0,…,4 are the reaction rate coefficients. k0,…,4 depend on the temperature following a similar relation as for dissociation, Eq. (). Table  shows the values of the reaction rate coefficients and their temperature dependence coefficients.

Equations () and () return the new concentration of SVI created in each SD in each time step. Based on the new concentration, the new mass of SVI and the new dry radius are calculated and the corresponding SD attributes are updated. The dry particle density of 1.8 g cm-3 is assumed while evaluating the dry radius from the SVI mass.

For the typical atmospheric conditions, say pH between 3 and 6 (i.e. [H+] between 10-3M and 10-6M), it can be said that the rate of oxidation by H2O2 depends very weakly on pH. In contrast, oxidation by ozone depends strongly on pH of the solution and can become very fast if pH is high. For example, increasing pH by 1 unit results in an approximately 100-fold increase in the O3 reaction rate.

The initial aerosol is assumed to be ammonium bisulfate (NH4HSO4), with a dry particle density of 1.8 g cm-3. Using the dry particle density and the dry radius of each SD, the initial mass of H+, NH4+ and SO42- ions is calculated. The initial mass of other molecules and ions is set to zero and is therefore not in equilibrium with the initial ambient trace gas conditions. For the initial conditions where supersaturation is present in the environment it is advisable to allow for a spin-up period with only condensation/evaporation and the equilibrium chemical processes enabled, to allow the model to reach equilibrium. Such initial conditions are mostly relevant for the kinematic models.

The kinematic model mimics a single 2-D eddy spanning a stratocumulus cloud deck and a boundary layer below it. The model is based on a test scenario from the 8th International Cloud Modeling Workshop ICMW;case 1. The velocity field is prescribed as in , , and . The same model was used when presenting the initial release of libcloudph++, see Sect. 2 in for details of the model formulation. The kinematic model is based on the open-source library of parallel MPDATA-based solvers for systems of generalized transport equations, see . The temperature, moisture, and trace gas fields are discretized on the Eulerian grid and are advected using the prescribed velocity field. Then, the model variables are passed to the PBMC scheme, where the microphysical and chemical processes are resolved. Finally, the source and sink terms due to microphysics and chemistry are calculated and applied in each model grid cell as described in Sects.  and .

The collisions between water drops are represented using the geometric kernel with collision efficiency for big drops (i.e. radius greater than 20 µm) from and for small droplets (i.e. radius smaller than 20 µm) from . For big drops, the collision efficiencies were obtained from the fit to measurements, see . For small droplets, the collision efficiencies were based on numerical simulations taking into account turbulence typical for stratocumulus clouds, see . The collision efficiencies are provided via a look-up table for different drop sizes.

The initial conditions are summarized in Table . The computational domain size is 1.5 km in both directions and the computational grid is composed of 75×75 cells of equal size (the grid lengths are 20 m) and is periodic in the horizontal direction. The initial air density profile corresponds to the hydrostatic equilibrium with the pressure of 1015 hPa at the bottom of the domain. At the beginning of the simulation it is assumed that there is no condensed water, and the initial profiles of θ and rv are constant with altitude. To keep the simulation set-up simple and due to a relatively low vertical extent of the computational domain, the initial trace gas volume fractions are also assumed to be constant with altitude. This unrealistic initial condition results in very high initial supersaturation in the upper part of the domain. As a consequence a 105 s (∼2 h 45 min) spin-up period is necessary to allow for the simulated water drops to reach equilibrium with their environment. During the spin-up only the reversible processes (condensation and evaporation, dissolving of trace gases, and dissociation into ions) are allowed and the supersaturation is limited to 5 % (relative humidity RH=1.05). After spin-up the simulations are run for 30 min. The chosen simulation time is enough to deplete the SO2 available in the cloudy part of the domain as well as to create precipitation.

Similar to the adiabatic parcel test, the initial aerosol is ammonium bisulfate and the aerosol particle size distribution is lognormal with one mode. The initial condition for trace gases is defined in terms of volume fractions and then translated to mixing ratios that serve as the model variables. The initial SO2, O3, and H2O2 volume fractions are taken from the simulation set-up used in . The values for SO2 and O3 are based on the measurements from the MASE campaign and the value for H2O2 is based on the representative values for the eastern Pacific Ocean . The NH3, HNO3, and CO2 volume fractions are the same as in the parcel test from Sect. .

The set-up detailed in Table  corresponds to “very clean conditions” (i.e. low aerosol particle concentrations). The initial aerosol particle sizes are also relatively small. Three additional simulation cases are studied to check the sensitivity of the model to different conditions. In case1 the reversible chemical processes are allowed, but the oxidation reaction is prohibited. In case2 the initial volume fraction of NH3 is increased and in case3 the initial aerosol size distribution is changed. The conditions for all the sensitivity simulation cases are summarized in Table .

As discussed in , the initial chemical scenario is idealized. For instance, although the initial conditions represent a clean maritime environment, the set-up lacks sea salt aerosol particles. As discussed by , sea salt aerosol particles are alkaline, which may in turn increase the pH of water drops and thus affect the oxidation rate. On the other hand, a study by indicates that alkaline sea salt particles are quickly converted to acidic due to the uptake of HCl vapour. More importantly, including sea salt would result in aerosol particles with very different hygroscopicity values κ of ammonium bisulfate is 0.61, whereas κ of NaCl is 1.28;. Including sea salt would also result in the initial bimodal size distribution with one mode representing smaller ammonium bisulfate aerosol particles and the second mode representing larger sea salt particles. In general, including sea salt should result in a very different condensational growth of aerosol particles. The set-up used in this study also lacks other particles containing sulfate, such as ammonium sulfate or sulfuric acid aerosol particles.

The initial aerosol size distribution parameters are based on the test cases studied in . The discussion presented in their study introduced two regimes for oxidation with regard to the mean aerosol size rd‾ and precipitation: (i) for a relatively small initial rd‾ production of sulfate enhances precipitation, and (ii) for a relatively big initial rd‾ production of sulfate suppresses precipitation. The overall impact depends strongly on the initial concentration of aerosol particles, see for the discussion. The short simulation time used in this study hinders analysis of the impact of oxidation on the overall precipitation. The work presented here focuses on the evolution of aerosol particle sizes and pH values of cloud and drizzle droplets. Future large eddy simulations should focus on the impacts of aqueous chemistry on precipitation, cloud lifetime, and cloud dynamics.

The kinematic set-up precludes any links between cloud microphysical processes and dynamics of the air motion. The set-up limits the study to the smooth velocity and therefore smooth saturation fields and prevents mixing between air parcels with different trajectories and properties. On the other hand, the kinematic set-up has low computational cost and allows for easy testing and sensitivity analysis. Prescribing the velocity ensures that all changes to the aerosol particle and water drop size distributions are caused by the cloud microphysics and aqueous-phase chemistry alone. Moreover, the kinematic set-up allows for a straightforward selection of the updraught and downdraught regions, further simplifying the analysis of the microphysical processes.

Figure  shows the model state after 30 min of simulation from the base case (see Table ). Figure a shows the concentration of the unactivated aerosol particles (defined as the SDs with wet radius smaller than 1 µm). The lower part of the plot (below 900 m) shows cloud-free conditions and corresponds to the initial concentration of aerosol particles. The upper part of the plot shows the interstitial aerosol particles, i.e. those in-cloud aerosol particles that did not activate. The difference between the upper and lower parts of Fig. a shows the impact of nucleation scavenging on aerosol population. The regions with slightly higher concentration of the in-cloud aerosol particles near the cloud base correspond to regions with low vertical velocities, lower supersaturations, and thus lower concentrations of the cloud droplets. Figure b shows the concentration of the cloud droplets (defined as the SDs with wet radii between 1 and 25 µm). It is nearly constant with height, that agrees with the observations in stratocumulus clouds e.g. . The regions with lower cloud droplet concentrations correspond to the regions with drizzle (see Fig. c). Figure c shows the rain water mixing ratio (water drops with wet radius greater than 25 µm) using a logarithmic colour scale. Rain forms quickly in the simulation due to the relatively high values of cloud droplet radii after the spin-up caused by the low initial aerosol particle concentration. The footprint of precipitation can be seen in Fig. b and f where the cloud droplet concentration is depleted in regions of drizzle. Figure d shows the mean dry radius of all particles (both the aerosol particles and water drops). The mean dry radius is increasing due to oxidation. In the updraught (left-hand side of panel d) the environmental aerosol particles that have not been affected by cloud are advected into the cloudy region. Once the cloud droplets are formed, the aqueous-phase oxidation starts to produce sulfate and changes the CCN size distribution. In the downdraught (right-hand side of panel d) cloud droplets are advected out of the cloud and they evaporate. The cloud-processed CCN are returned to the environment and change the ambient air aerosol particle size distribution. Figure e depicts the cloud droplet effective radius. As expected, the effective radius increases with height. At the top of the cloud the effective radius reaches 20 µm, which is linked to the small cloud droplet concentration. High effective radii imply efficient drizzle production after the spin-up (usually water drop radius ∼12 µm is reported as the threshold value for efficient collisions between water drops and the production of precipitation, for example ). Figure f shows the concentration of SVI molecules (all molecules containing sulfur at +VI oxidation state) and represents molecules from the initial ammonium bisulfate (NH4HSO4) aerosol and the molecules created during oxidation. It corresponds to the mean dry radius plotted in Fig. d. Additionally, some effects of collisions and precipitation can be seen when comparing the irregular features from Fig. f with rain water mixing ratio in Fig. c. Precipitation displaces the largest water drops, which causes the irregular distribution of SVI molecules in cloudy grid cells. Figure f also shows that the particle-based scheme can track the dissolved chemical compounds in the evaporating rain drops below the cloud base.

Figure b, d, e, and f show a layer of very clean air above the cloud which is caused by sedimentation of cloud droplets. In the downdraught region, the prescribed velocity field advects the clean layer into the domain. This feature is not present in the aerosol concentration plot (Fig. a) because the clean layer contains small aerosol particles with small sedimentation velocity. The depicted clean layer is an artifact caused by the prescribed velocity field and the absence of aerosol sources in the computational domain. The relatively short simulation time is chosen to minimize the impact of the clean layer on the simulation.

Figure  shows the liquid water volume weighted average pH in each computational grid cell from the base case (a) and sensitivity test cases (b–d). In order to better adjust the colour scale to the in-cloud pH variability, pH values below 3 that correspond to very acidic aerosol particles below the cloud base have been clipped. Figure  captures the pH of cloud droplets as well as the pH of some evaporating rain drops below the cloud base. The droplets in the downdraught of Fig. a are more acidic due to SVI created during aqueous-phase oxidation. For the base case, Fig. a, pH increases with height above the cloud base. Initially the water drops are very acidic, but as they grow in size they become more diluted. Even though SVI is created during oxidation, the average pH still increases with height due to dilution. The same behaviour is shown in the adiabatic parcel tests discussed in Sect.  and shown in Fig. . The increase in pH with height is also observed in a 1-dimensional model representing processing of sulfur in small cumuli in marine environment used by . Due to the pH variability shown in Fig. a, oxidation by O3 happens mostly near the cloud top in the base case. As discussed in Sect. , the rate of oxidation by O3 increases significantly with increasing pH (see Eq. ), whereas oxidation by H2O2 depends very weakly on the acidity (Eq. ). The study by also reported that the pH of droplets increased with height due to dilution despite the production of sulfuric acid. In turn, increased pH promotes oxidation by O3 in the upper parts of the cloud, whereas oxidation by H2O2 dominates in lower parts of the cloud, according to their study.

Case1 shown in Fig. b represents a hypothetical “no oxidation” scenario where all physical and chemical conditions are the same as in the base case, the reversible chemical processes are allowed, and the oxidation reaction is prohibited. The scenario without oxidation is overall less acidic than the base case (Fig. a). Additionally, without oxidation there is no difference between the pH values in the updraught and downdraught in Fig. b. Without oxidation, all the chemical processes are reversible and the dissolved chemical compounds are outgassed to the atmosphere as the cloud droplets evaporate in the downdraught.

Case2 differs from the base case by increasing the initial NH3 volume fraction from 0.1 to 0.4 ppbv (see Tables  and ). Because the initial aerosol particle size distribution is the same as in the base case, the mean aerosol and droplet sizes and concentrations at the end of the simulation are not different from the base case (not shown). Figure c shows the liquid water volume weighted average pH for case2. The average pH in case2 (Fig. c) is higher than in the base case (Fig. a), that is, both cloud droplets and rain drops are less acidic in case2 than in the base case. In contrast to the base case, in the updraught (left-hand side of the plots), the pH in case2 actually decreases with height above the cloud base. This is because the higher initial NH3 volume fraction increases its uptake and counters the low pH values caused by the initial acidic aerosol particles, see Eq. (). Then, as the water drops are advected upwards, oxidation produces sulfuric acid and the average pH decreases. Near the cloud top, the NH3 is degassed back to the environment. The case2 results are in agreement with the trajectory ensemble model simulations by . In their study, the initial aerosol size distribution is the same as in the base case and case2. However, their initial trace gas volume fractions are much higher and aim to represent a “moderately polluted marine environment” (their base case NH3 volume fraction is 10 times larger than the base case value assumed here). As in case2 presented here, the high initial NH3 volume fractions in increase the pH near the cloud base and promote oxidation by O3 during the first minutes after the simulated parcels entered the cloud. Because the sulfuric acid was produced, the pH dropped and oxidation by H2O2 becomes dominant in the higher regions of the cloud, as reported in their study.

Case3 increases the initial aerosol concentration to 150 cm-3, while keeping all other initial conditions the same as in the base case (see Tables  and ). In general, higher initial aerosol particle concentration results in higher cloud droplet concentrations. This in turn creates smaller cloud droplet effective radii that virtually prohibits the onset of precipitation during the 30 min simulation time (not shown). Figure d shows the liquid water volume weighted average pH for case3. Similar to the base case (Fig. a), the pH increases with height due to the dilution and the downdraught droplets are more acidic due to the ongoing oxidation. However, case3 is more acidic than the base case because the overall droplet sizes are smaller and they are therefore less diluted.

At the end of the base case simulation, 18 % of the total available SIV is oxidized. As a result, 0.14 µg m3 of the dry particulate matter is created during oxidation per cubic metre of air (an average value for the whole computational domain reported in relation to the dry air volume). In total, 40 % of the final dry particulate matter is created due to oxidation and 60 % originates from the initial aerosol mass. The oxidation is a significant source of the dry particulate matter because the initial aerosol mass is very low (only 0.21 µg m3 dry air). Oxidation by H2O2 is the dominant path: 92 % of SVI originates from SIV oxidation by H2O2. More alkaline conditions of case2 enhance the efficiency of oxidation. At the end of case2, 21 % of available SIV is oxidized. As a result, oxidation produces 0.16 µg of dry particulate matter per cubic metre of dry air (average over the whole computational domain). For case2, 44 % of the final dry particulate matter is created due to oxidation and 56 % originates from the initial ammonium bisulfate aerosol. Similar to the base case, the significance of oxidation as a source of dry particulate matter is caused by a very low initial aerosol mass. Due to more alkaline conditions, oxidation by O3 becomes more important than in the base case. At the end of the case3 simulation, 39 % of the SVI originates from SVI oxidation by O3 and 61 % by H2O2. In contrast, more acidic conditions of case3 hinder the O3 reaction path. Virtually all molecules of sulfate that are created during oxidation are oxidized by H2O2. As a result, the conversion of sulfur to sulfate is slightly less effective in case3. At the end of the case3 simulation, 17 % of available SIV is oxidized. As a result, 0.13 µg of dry particulate matter is created per cubic metre of dry air. At the end of case3 simulation, 17 % of the dry particulate matter is created by oxidation and 83 % originates from the initial aerosol. The initial aerosol mass is larger in case3 than in the base case due to the higher initial aerosol concentration (case3 contains initially 0.61 µg m3 of dry particulate matter). Due to the simple kinematic set-up chosen in this study the values reported here cannot be treated as representative of the atmospheric conditions. They are shown to allow comparison between the base case and the sensitivity test cases.

Finally, the impact of collisions and aqueous-phase oxidation of sulfur on the aerosol and water drop size distributions is examined. For this purpose, the aerosol particle size distributions from the base case (Fig. a) and case3 (Fig. b) are compared. The black line represents the initial aerosol size distribution, and the green and red lines represent the final aerosol size distribution for the in-cloud (rc>0.01 g kg-1) and precipitating (rr>0.01 g kg-1) grid cells, respectively. The two cases are chosen because they have different initial aerosol size distributions. In both cases the cloud-processed aerosol size distributions (green and red lines) have a bimodal shape. This is a footprint of oxidation that creates the Hoppel minimum in the dry radius size distribution. The same effect is obtained in the adiabatic parcel tests discussed in Sect.  and shown in Fig. . Moreover, the efficient collisions between water drops in the base case create a tail of bigger aerosol sizes in Fig. a. The effect is stronger for the precipitating grid cells (red line). In case3 fewer collisions between water drops occur than in the base case and therefore no precipitation and no tail of big aerosol particles is created. Also, in case3, the change in size distribution of aerosol particles caused by oxidation is smaller because the produced sulfate is divided among a larger number of aerosol particles.